Citation for this version held on GALA:Haider, Jibran, Lee, Chun Hean, Gil, Antonio J. and Bonet, Javier (2016) A first order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme. London: Greenwich Academic Literature Archive. , a first order hyperbolic system of conservation laws was introduced in terms of the linear momentum and the deformation gradient tensor of the system, leading to excellent behaviour in two dimensional bending dominated nearly incompressible scenarios. The main aim of this paper is the extension of this algorithm into three dimensions, its tailor-made implementation into OpenFOAM and the enhancement of the formulation with three key novelties. First, the introduction of two different strategies in order to ensure the satisfaction of the underlying involutions of the system, that is, that the deformation gradient tensor must be curl-free throughout the deformation process. Second, the use of a discrete angular momentum projection algorithm and a monolithic Total Variation Diminishing Runge-Kutta time integrator combined in order to guarantee the conservation of angular momentum. Third, and for comparison purposes, an adapted Total Lagrangian version of the Hyperelastic-GLACE nodal scheme of Kluth and Després [2] is presented. A series of challenging numerical examples are examined in order to assess the robustness and accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies developed by the authors in recent publications.

The paper presents a new computational framework for the numerical simulation of fast large strain solid dynamics, with particular emphasis on the treatment of near incompressibility. A complete set of first order hyperbolic conservation equations expressed in terms of the linear momentum and the minors of the deformation (namely the deformation gradient, its co-factor and its Jacobian), in conjunction with a polyconvex nearly incompressible constitutive law, is presented. Taking advantage of this elegant formalism, alternative implementations in terms of entropy-conjugate variables are also possible, through suitable symmetrisation of the original system of conservation variables. From the spatial discretisation standpoint, modern Computational Fluid Dynamics code "OpenFOAM" [http://www.openfoam.com/] is here adapted to the field of solid mechanics, with the aim to bridge the gap between computational fluid and solid dynamics. A cell centred finite volume algorithm is employed and suitably adapted. Naturally, discontinuity of the conservation variables across control volume interfaces leads to a Riemann problem, whose resolution requires special attention when attempting to model materials with predominant nearly incompressible behaviour (κ/µ ≥ 500). For this reason, an acoustic Riemann solver combined with a preconditioning procedure is introduced. In addition, a global a posteriori angular momentum projection procedure proposed in [1] is also presented and adapted to a Total Lagrangian version of the nodal scheme of Kluth and Després [2] used in this paper for comparison purposes. Finally, a series of challenging numerical examples is examined in order to assess the robustness and applicability of the proposed methodology with an eye on large scale simulation in future works.

This paper presents a new Smooth Particle Hydrodynamics computational framework for the solution of inviscid free surface flow problems. The formulation is based on the Total Lagrangian description of a system of first-order conservation laws written in terms of the linear momentum and the Jacobian of the deformation. One of the aims of this paper is to explore the use of Total Lagrangian description in the case of large deformations but without topological changes. In this case, the evaluation of spatial integrals is carried out with respect to the initial undeformed configuration, yielding an extremely efficient formulation where the need for continuous particle neighbouring search is completely circumvented. To guarantee stability from the SPH discretisation point of view, consistently derived Riemann-based numerical dissipation is suitably introduced where global numerical entropy production is demonstrated via a novel technique in terms of the time rate of the Hamiltonian of the system. Since the kernel derivatives presented in this work are fixed in the reference configuration, the non-physical clumping mechanism is completely removed. To fulfil conservation of the global angular momentum, a posteriori (least-squares) projection procedure is introduced. Finally, a wide spectrum of dedicated prototype problems is thoroughly examined. Through these tests, the SPH methodology overcomes by construction a number of persistent numerical drawbacks (e.g. hour-glassing, pressure instability, global conservation and/or completeness issues) commonly found in SPH literature, without resorting to the use of any ad-hoc user-defined artificial stabilisation parameters. Crucially, the overall SPH algorithm yields equal second order of convergence for both velocities and pressure.

This article presents a vertex-centered finite volume algorithm for the explicit dynamic analysis of large strain contact problems. The methodology exploits the use of a system of first order conservation equations written in terms of the linear momentum and a triplet of geometric deformation measures (comprising the deformation gradient tensor, its co-factor, and its Jacobian) together with their associated jump conditions. The latter can be used to derive several dynamicconservation laws, explicit contact dynamics, large strain, OpenFOAM, Riemann solver, shocksThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

In practical engineering applications involving extremely complex geometries, meshing typically constitutes a large portion of the overall design and analysis time. In the computational mechanics community, the ability to perform calculations on tetrahedral meshes has become increasingly important. For these reasons, automated tetrahedral mesh generation by means of Delaunay and advancing front techniques have recently received increasing attention in a number of applications, namely: crash impact simulations, cardiovascular modelling, blast and fracture modelling. Unfortunately, modern industry codes in solid mechanics typically rely on the use of traditional displacement based Finite Element formulations which possess several distinct disadvantages, namely: (1) reduced order of convergence for strains and stresses in comparison with displacements; (2) high frequency noise in the vicinity of shocks; and (3) numerical instabilities associated with shear locking, volumetric locking and pressure checker-boarding. In order to address the above mentioned shortcomings, a new mixed-based set of equations for solid dynamics formulated in a system of first order hyperbolic conservation laws was introduced. Crucially, the new set of conservation laws has a similar structure to that of the well known Euler equations in the context of Computational Fluid Dynamics (CFD). This enables us to borrow some of the available CFD technologies and to adapt the method in the context of solid dynamics. This thesis builds on the work carried out by Lee et al. 2013 by further developing the upwind cell centred finite volume framework for the numerical analysis of large strain explicit solid dynamics and its tailor-made implementation within the open source code OpenFOAM, extensively used in industrial and academic environments. The object oriented nature of OpenFOAM implementation provides a very efficient platform for future development. In this computational framework, the primary unknown variables are linear momentum and deformation gradient tensor of the system. Moreover, the formulation is further extended for an additional set of geometric strain measures comprising of the co-factor of deformation gradient tensor and the Jacobian of deformation, in order to simulate polyconvex constitutive models ensuring material stability. The domain is spatially discretised using a standard Godunov-type cell centred framework where second order accuracy is achieved by employing a linear reconstruction procedure in conjunction with a slope limiter. This leads to discontinuities in variables at the cell interface which motivate the use of a Riemann solver by introducing an upwind bias into the evaluation of numerical contact fluxes. The acoustic Riemann solver presented is further developed by applying preconditioned dissipation to improve its performance in the near incompressibility regime and extending its range to contact applications. Moreover, two evolutionary frameworks are proposed in this study to satisfy the underlying involutions (or compatibility conditions) of the system. Additionally, the spatial discretisation is alternatively represented through a nodal cell centred finite volume framework for comparison purposes. From a temporal discretisation point of view, a two stage Total Variation Diminishing Runge-Kutta time integrator is employed to ensure second order accuracy. Additionally, inclusion of a global posteriori angular momentum projection procedure enables preservation of angular momenta of the system. Finally, benchmark numerical examples are simulated to demonstrate various aspects of the formulation including mesh convergence, momentum preservation and the locking-free nature of the formulation on complex computational domains. En aplicaciones prácticas de ingeniería que implican geometrías extremadamente complejas, el mallado requiere típicamente una gran parte del tiempo total de diseño y análisis. En la comunidad de mecánica computacional, la capacidad de realizar cálculos sobre mallas tetraédricas está siendo cada vez más importante. Por estas razones, la generación automatizada de mallas tetraédricas por medio de técnicas de Delaunay y frente avanzado han recibido cada vez más atención en ciertas aplicaciones, a saber: simulaciones de impacto, modelado cardiovascular, modelado de explosión y fractura. Por desgracia, los códigos en la industria moderna para mecánica de sólidos se basan normalmente en el uso de formulaciones tradicionales de Elementos Finitos formulados en desplazamientos que poseen varias desventajas: (1) menor orden de convergencia para tensiones y deformaciones; (2) ruido de alta frecuencia cerca de las ondas de choque; y (3) inestabilidades numéricas asociadas con el bloqueo a cortante, el bloqueo volumétrico y oscilaciones de presión. Con el fin de abordar estas deficiencias, se introduce un nuevo conjunto de ecuaciones para mecánica del sólido formulada como un sistema de leyes de conservación de primer orden basada en una formulación mixta. Fundamentalmente, el nuevo sistema de leyes de conservación tiene una estructura similar a la de las famosas ecuaciones de Euler en el contexto de la Dinámica de Fluidos Computacional (CFD). Esto nos permite aprovechar algunas de las tecnologías CFD disponibles y adaptar el método en el contexto de la Mecánica de Sólidos. Esta tesis se basa en el trabajo realizado en Lee et al. 2013 mediante el desarrollo de la estructura de volúmenes finitos centrados en celdas upwind para el análisis numérico de dinámica del sólido explícita en grandes deformaciones y su implementación específicamente diseñada dentro del software de código abierto OpenFOAM, ampliamente utilizado ámbito académico e industrial. Además, la naturaleza orientada a objetos de su implementación proporciona una plataforma muy eficiente para su desarrollo posterior. En este marco computacional, las incógnitas básicas de este sistema son el momento lineal y el tensor gradiente de deformación. Asimismo, la formulación se extiende adicionalmente para un conjunto adicional de medidas de deformación que comprenden el cofactor del tensor gradiente de deformación y el jacobiano de deformación, con el fin de simular modelos constitutivos policonvexos que aseguran la estabilidad del material. El dominio se discretiza espacialmente usando un marco centrado en células de tipo Godunov estándar, donde se consigue la precisión de segundo orden empleando un procedimiento de reconstrucción lineal junto con un limitador de pendiente. Esto conduce a discontinuidades en las variables en la interfase de la célula que motivan el uso de un solucionador de Riemann mediante la introducción de un sesgo contra el viento en la evaluación de flujos de contacto numéricos. El presente solucionador acústico de Riemann es posteriormente desarrollado aplicando disipación pre-condicionada para mejorar su rendimiento en el cercano pero incompresibilidad régimen y extender su gama a aplicaciones de contacto. Además, se proponen dos marcos evolutivos en este estudio para satisfacer las involuciones subyacentes (o condiciones de compatibilidad) del sistema. Además, la discretización espacial se representa alternativamente a través de un marco de volumen finito centrado en células nodales para fines de comparación. Desde el punto de vista de la discretización temporal, se emplea un integrador temporal de Runge-Kutta de dos etapas con Disminución de Variación Total para asegurar segundo orden de precision. Finalmente, se simulan ejemplos numéricos de referencia para demostrar varios aspectos de la formulación que incluyen convergencia de malla, conservación de momento y la naturaleza libre de bloqueo de la formulación en dominios computacionales complejos.